# HW7 - the receiver of the message uses “majority”...

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OR 3500/5500, Summer’08, Chen Homework 7 Due on Monday, June 15, 3pm. Problem 1 Suppose Charles is taking an exam consisting of 100 problems and the proba- bility of getting a particular question right is 1 / 2. Let X denote the number of questions he gets right. (a) Using Central Limit Theorem, ﬁnd a and b such that aX + b is approximately N (0 , 1). (b) Approximate the probability of his passing (recall he needs 40 right to pass) by an integral of the standard normal density. Calculate the integral using some software package. Problem 2 X is exponentially distributed with parameter λ . If P ( X > 0 . 01) = 1 / 2, ﬁnd the unique number t such that P ( X > t ) = 0 . 9 . Problem 3 A communication channel transmits digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability . 2. Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit 00000 instead of 0 and 11111 instead of 1. If
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Unformatted text preview: the receiver of the message uses “majority” decoding, what is the probability that the message will be incorrectly decoded? What independence assumption are you making? (By majority decoding we mean that the message is decoded as “0” if there are at least 3 zeros in the message received and as “1” otherwise.) Problem 4 Let X be a binomial random variable with parameters n and p . (a) Show that P ( X = k + 1) = p 1-p n-k k + 1 P ( X = k ) , k = 0 , 1 ,...,n-1 . (b) As k goes from 0 to n , P ( X = k ) ﬁrst increases and then decreases. Show that this probability reaches its largest value when k is the largest integer less than or equal to ( n + 1) p . The following problem is optional for 3500 students, mandatory for students enrolled in 5500 Problem 5 If X has a Poisson( λ ) distribution, then ﬁnd E [ X ( X-1)( X-2) ··· ( X-k )]. 1...
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## This note was uploaded on 09/22/2009 for the course ORIE 3500 taught by Professor Weber during the Summer '08 term at Cornell.

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